 In this unit, we will develop the graphs for y = sin(x),
y = cos(x), and later y = tan(x), and identify the
characteristic features of each.
 Transformations of y = sin(x) and y = cos(x) are
performed and are used to model and solve problems.
 You are not expected to transform y = tan(x) or solve
problems that involve the tangent function.
Going Around in Circles
 Mathematical Models for Rotating Wheels
 Ferris Wheels
 Car Wheels
 Here we look at the relationship between time and the
height of a point on the outside of a rotating wheel.
Ferris Wheels
 Where do you start your ride when you get on a Ferris
Wheel?
 You start at the bottom.
 As you begin to move, you start to rise above the
ground so you are increasing your height.
 Your max height is reached when you reach the top
and then your height gets less until you reach your
minimum height at the bottom of the ride.
 Thus your height is constantly changing as time goes
on.
Graph of Your Ferris Wheel Ride!
 What is the Max Height? ________
 What is the min height? ________
 How long does it take to make one complete
revolution? _________
Components of the Graph
NOTE: This graph is a SINUSOIDAL Graph
Introduction to Sinusoidal
Functions
 There are many examples in the real world that can be
modelled by Sinusoidal Functions.
 Wheels rotating.
 Ocean waves.

Important to surfers
 Tides
 Hours of Daylight
 Electrical Current

Alternating Current
Periodic Functions
 Data that repeats itself is called periodic and is
represented by a periodic function.
 Heart Beats
 The amount of time (distance between corresponding
x-values) it takes for the data (y –values) to repeat is
called the period.
 What is the period for this graph?
 The main type of periodic function studied in this unit
is sinusoidal.
 Sinusoidal functions have a wave shape repeating in
horizontal direction.
 Similar to the Ferris wheel graph.
Describe the following graphs as
Periodic and/or sinusoidal.
Components of a Sinusoidal Graph
Period
 Local Max
Amplitude
 Largest y- value
 Local Min
 Smallest y-value
 Period
 Horizontal distance required for graph to repeat itself
 Sinusoidal Axis
 Centre line of graph
 Amplitude
 Distance from sinusoidal axis to max( or min) point
SA
Graph each function for the domain 0 ≤ θ ≤ 3π.
a) y = sin θ b) y = cos θ
A) Complete the table of values. Round values to one decimal
place.
B) Plot the points and join
them with a smooth curve.
b) y = cos θ
5 key points
 There's 5 key points we associate with the graphs, since
these points will help us determine the characteristics
of the graph (amplitude, domain, range, period, zeros,
and sinusoidal axis).
 However we are not limited to these points.
Graph of y = sin x
 y-intercept:
 x-intercepts(zeros):
 max value:
 min value:
 Amplitude
 Sinusoidal Axis  max  min
Period
 Domain
Range min, max
2

max  min
2
Graph of y = cos x
 y-intercept:
 x-intercepts(zeros):
 max value:
 min value:
 Amplitude
 Sinusoidal Axis  max  min
Period
 Domain
Range min, max
2

max  min
2
Transformations of Sinusoidal
Functions
 In Unit 2 we worked with transformations of graphs in
the form y = af (b(x − h)) + k .
 We will now apply that knowledge to sinusoidal graphs:
 y = a sin b (x − c )+ d or y = a cos b (x − c )+ d
 Note: Parameters h and k are now referred to as c and d.
 We will determine the end position of the five key
points using transformations (mapping rule), and then
extend the graph appropriately.
 We need to know what characteristics of the graph
change with each parameter.
 To achieve this, we will investigate how each parameter
change affects the resulting graph one parameter at a
time, and match these changes to the defining
characteristics of sinusoidal graphs.
1. Determine how varying the value of a affects
the graph of y = a sin x and y = a cos x.
 The value of “a” is a Vertical Stretch (and/or
Reflection)
 Sketch the graphs of:
A) y = 2sinx

B)
y= 0.5sinx
y





y
x










x










 What characteristic(s) of the sinusoidal graph changed
with “a”?
 Amplitude
 Max and min values
 Range
 For the function y = a sin x, (and y = a cos x) the
amplitude is |a|.
C) y = -3cos x
What is the amplitude?

y


x










What is the Range?
Note: There is also a vertical reflection about the
x-axis
Example: Determine the amplitude
of each function.
A) y = -4sin x
B) -2y= cos x
C) 5y=40 sin x
2. Determine how varying the value of b affects
the graph of y = sin bx and y = cos bx.
 The value of “b” is a Horizontal Stretch (and/or
Reflection)
 Sketch the graphs of:
A) y = sin(2x)

B)
y= sin (0.5x)
y


y

x





x









 What characteristic(s) of the sinusoidal graph
changed with “b”?
 Period
 For the function y = sin bx, (and y = cos bx) the new
period is
new period 
Radians
2
new period 
|b |
old period
|b |
Degrees
360o
new period 
|b |
C) y = cos -4x
What is the period?

y


x










Note: There is also a Horizontal reflection about the
y-axis.
However, in this case there is no change since cosine is
symmetric about the y-axis
Example: Determine the amplitude
of each function.
 A) y = -4sin x
B) -2y= cos 3x
C)
y=4 sin (0.1x)
Example: Sketch the graph of y = ‐3sin2x
for at least two cycles.
 Determine the amplitude, period, max and min values, x‐intercepts, y‐
intercepts, domain and range and equation of sinusoidal axis

y
y=sin x
(x, y)


x














y = ‐3sin2x
Example: Sketch the graph of y = ‐3sin2x
for at least two cycles.
 Amplitude: ____

y
 Period: ____
 max and min values:


 x‐intercepts:
x y‐ intercepts:














Domain ______
 Range ________
 Equation of sinusoidal axis.
Page 233 -235
# 6 – 10, 14
Transformations of Sinusoidal
Functions
1. Determine how varying the value of d affects
the graph of y = sin x + d and y = cos x + d
 The value of “d” is a Vertical Translation
 Sketch the graphs of:
A) y = sin x + 1

B) y= sin x - 2
y





y
x










x










 What characteristic(s) of the sinusoidal graph changed
with “d”?
 Sinusoidal Axis
 Max and min values
 Range
 For the function y = sin x + d, (and y = cos x + d) the
equation of the Sinusoidal Axis is y = d.
C) y = cos x + 1.5

y


x










What is the equation of the sinusoidal axis?
What is the range?
Example: Determine the equation of
the sinusoidal axis of each function.
A) y = sin x +4
B) y - 2= cos x
C) 5y= 5sin x-40
What is the equation of the
function below?

y


x














2. Determine how varying the value of c affects
the graph of y = sin (x – c) and y = cos (x – c)
 The value of “c” is a Horizontal Translation
 Sketch the graphs of:


A) y = sin (x – 2 )

B) y= sin (x + 2 )
y





y
x










x










 What characteristic(s) of the sinusoidal graph changed
with “c”?
 Starting point (middle of upward slope for sine)
 With sinusoidal functions the horizontal translation is
referred to as the phase shift, and will determine how
the 5 key points will be translated horizontally
 For the function y = sin(x – c), (and y = cos (x – c)) c is
the phase shift.
C) y = cos (x - 2 )

y


x










What does y = cos (x - 2 ) equal?
Example: Determine the phase shift of
each function.
A) y = sin (x + 4)
B) y = cos (x -  ) C) y= sin (4x-6  )
Example: Match each function with
its corresponding function rule

y



y


















x


y
x
x












y












x




















Page 250
#1 a,b, 2a,b, 5
y = a sin b (x − c )+ d OR y = a cos b (x − c )+ d
Mapping Rule:
Parent Graph
x
y
0o
0
90o
1
180o
0
270o
-1
360o
0
Period = _____
Amplitude = ____
Sinusoidal Axis: ______
Max y value =_____
Min y value = _____
y  sin x
Range: ____________
Domain: ___________
OR in Radians
 Period = _____
 Amplitude = ____
 Sinusoidal Axis: ______
 Max y value =_____
 Min y value = _____
 Range: ____________
 Domain: ___________
Examples: Graph the following
( y  2)  sin( x  45 )
 Mapping rule:
__________________________
(x,y)
 a: _____
 b: _____
 c: _____
 d: _____
Period:______
Amplitude:______
Sinusoidal Axis:______
2. y  2cos(x  30)  1
 Mapping rule:
__________________________
(x,y)




a: _____
b: _____
c: _____
d: _____
Period:______
Amplitude:______
Sinusoidal Axis:______
Starting Point: _______
3
3. 2(y  1)  cos(x  )
4

 Mapping rule:
__________________________
y

(x,y)




a: _____
b: _____
c: _____
d: _____

x












Period:______
Amplitude:______
Sinusoidal Axis:______
Starting Point: _______

max y value:__________
min y value: ___________ Range: ___________

1
1
4. (y  3)  sin (x  2)
3
2





 Mapping rule:
__________________________
 a: _____
(x,y)
 b: _____
 c: _____
 d: _____
Period:______
Amplitude:______
Sinusoidal Axis:______
Starting Point: _______
Jump: ________

y








max y value:__________
min y value: ___________
x









y = a sin b (x − c )+ d
d  a
Summary
c
 Eq of Sinusoidal Axis:
 Amplitude:
 Local Max:
 Local Min:
 Period:
 Start:
a
period
4
y d
c
c
(sin)
d a
360o
2
or
b
b
y d
a
 Jump:
 Range: min, max
(cos)
360o
2
or
b
b
d a
period
4
d  a
Page 250
#3
1
1. Reading a Sinusoidal Equation 5 (y  6)  sin2(x  135)
y
y
Reflection?______
Vertical Stretch:______
Vertical Translation:______
Horizontal Stretch:______
Horizontal Translation:______
Amplitude:______
Sinusoidal Axis________
Period:______
Starting Point:______
Max Value:_______
Min Value:_______
Range: _________
Sketch Graph













x















Example 2
1
2
y  2cos (x  )  11
3
3
y


Reflection?______
Vertical Stretch:______
Vertical Translation:______
Horizontal Stretch:______
Horizontal Translation:______
Amplitude:______
Sinusoidal Axis________
Period:______
Starting Point:______
Max Value:_______
Min Value:_______











x
Sketch Graph









Example 3.
2
 y  sin( 4x  12)  2
3
Reflection?______
Vertical Stretch:______
Vertical Translation:______
Horizontal Stretch:______
Horizontal Translation:______
Amplitude:______
Sinusoidal Axis________
Period:______
Starting Point:______
Max Value:_______
Min Value:_______
Range: __________
y





x

Sketch Graph


4. Write an equation for each transformation of y = sin x given the
mapping rules below:
A ) (x , y )  (x  2, y  3)
B ) (x , y )  (x  4,3y  1)
1

C ) (x , y )   x , 4y 
3

2


D ) (x , y )   5x  120, y  11 
7


Page 250
#4
1. Find the equation of the graph below as a transformation of y = sinx.
Formulae
max  min
 Amplitude _______ VS = a 
2
max  min
 Sinusoidal Axis:_______VT = d 
2
We need to find the values
of a, b, c, and d for the
equation : y = a sin b (x − c )+ d
o
2
2

360
new period 
| b |
or | b |
 Period:_______
|b |
new period
new period
 Starting Point: ______ phase shift =HT  c
There are an infinite number of
correct choices for the phase shift, the
convention is to use the smallest positive
value.
 NOTE:
Equation:
Bonus: What is the equation if we used
y = cos x?
2.
Find the equation of the graph below
 Amplitude_______
 Sinusoidal Axis:________
 Period:________
 Starting Point: ______
Equation:___________________
3. Find the equation of the graph that models the height
of the piece of tape as the can rolls across the floor.
y
0 cm
15
12
9
6
3
Tape
5 10 15 20
x
Distance Rolled(cm)
Equation:
BONUS: What is the diameter of the can?
SUPER-DUPER Bonus: What is the height of the tape when the can rolls 42 cm?
Multiple Choice Questions…
1.
If the amplitude of a
sinusoidal function is 3, and
the maximum point is 4,
what is the equation of the
sinusoidal axis?
2. If the equation of the
sinusoidal axis is y = 1 and
the maximum point of a
sinusoidal function is 4,
what is the minimum point
on the graph?
A. y = 7
y=1
C. y = -1
D. y = -2
B.
A. - 4
B. - 3
C. - 2
D.
7
Multiple Choice Questions…
3.
If the starting point of a
sine wave is -45° and the
next cycle begins at 315°,
what is the period of the
function in degrees?
4. If a maximum point of a
sinusoidal function is 1 and
a minimum point is - 6,
what is the equation of the
sinusoidal axis?
A. 180
A. y = -5
270
C. 315
D. 360
B. y = -5/2
B.
C. y = 7
D. y = 7/2
For the following multiple choice questions, use the equation:
2 ( y - 5 ) = sin 3 ( x + 180o )
5.
What is the amplitude of the
above equation?
7. What is the maximum point on
A.
B.
C.
D.
½
1
2
5
A. 4.5
B. 5
C. 5.5
D. 7
6. What is the period of the above
function, in degrees?
A.
B.
C.
D.
120
180
360
1080
the graph?
8. What is the range of the
function?
A. {y/ -5 < y < 5}
B. {y/ 3 < y < 7}
C. {y/ -7 < y < 7}
D. {y/ 4.5 < y < 5.5}
Multiple Choice Questions…
9.
If the horizontal stretch factor
of a sinusoidal function is 3,
what is the period of its graph,
in degrees?
10. What is the horizontal stretch
factor of a sinusoidal function
whose period is 90 degrees?
¼
B. ½
C. 2
D. 4
A.
60
B. 120
C. 540
D. 1080
A.
Page 251-3
#6,7,8,9, 14-16
1.
A pebble is stuck in the grooves of a car tire. As the car moves at
a speed of 60 inches/sec, the height of the pebble changes. If the tire
has a 15-inch radius, graph the path traveled by the pebble. Determine
the equation of the graph.
2. Jackie, Nicole and Megan are playing skip rope. As the rope rotates it is
observed that its maximum height is 3m and its first maximum occurs 1s
after starting. If the first minimum occurs 2s after the maximum at a
height of 0.2m. Determine the equation that expresses the skip ropes
height above the ground in terms of time
3. A mother puts her child on a Merry-Go-Round. To watch her child, she
stands at a point that is initially 3 metres away from the child, which is the
closest distance between the mother and child. At 6 seconds, the child is
15 metres from his mother, which is the farthest distance between the
mother and child. Assuming the distance between the mother and child
varies sinusoidally with time, determine the relation that models this
situation.
distance (m)
y









x





time (seconds)
4. Determine the equation that models the
hours of daylight for one complete year in
Corner Brook, NL
 We need some data.
http://www.timeanddate.com/worldclock/sunrise.html
 When is the shortest day?
 Dec 21
 What is the length of the shortest day? 8 hours 14 min

8.23 h
 What day of the year is this? 355
 When is the longest day?
 Jun 21
 What is the length of the longest day? 16 hours 12 min

16.37 h
 What day of the year is this? 172
 What is the average daylight hours? 12. 3 hours
 What is the equation?
 What is the hours of daylight for today?
Page 253-5
#20,26A, 27A) C)
 What is the slope of the terminal
arm for angle  on the unit circle
as it rotates from 0° to 90°?
 Lets start with  at 0o
slope  m 
y 0
rise
  0
run
x 1
 What happens as the angle
increases?
r=1
y
x
 What is the slope at 10o?
y
sin 
sin10o 0.1736
slope 


 0.1763

o
x
cos 
cos10 0.9848
 What is a quicker way to
calculate the slope?
m
sin 
 tan 
cos 
tan10 
 Find the slope of the terminal arm
for the following values of .

30o
45o
60o
75o
89o
89.9o
Slope
 What happens to the slope as the
angle approaches 90°?
r=1
 Slope approaches infinity.
y
 What is the slope of the line at 90°?
 Undefined ( ∞ )
 Which other angles would provide
x
the same slopes as those above?

Other
Angles
30o
45o
60o
75o
89o
89.9o
 What are the slopes for rotations
from 0° to −90°?

0o
-30o
-45o
-75o
-89o
-90o
Slope
 Compare these to slopes when the
terminal arm is rotated past 90°.

r=1
90o
91o
105o
135o
150o
y
Slope
x
 Is the tangent function periodic?
 Yes.
 What is its period?
 180o or π radians
180o
For the function: y = tan x
 Recall: tan x  sin x
cos x
 Where is tan x undefined?
 Places where cos x = 0
x 

2
 k , k  Z
Plot these as
Vertical Asymptotes
 Note: An Asymptote is a line that a graph approaches, but
never reaches.
 There are:
 Vertical Asymptotes where a function’s y values approach infinity
 Horizontal Asymptotes where a function goes to a certain y-value
as the x values approach infinity (or negative infinity)
 Slant Asymptotes where a function approaches a line as the x values
approach infinity (or negative infinity)
 Where would tan x cross the x- axis? (x- intercepts)
 Places where sin x = 0
x  k , k  Z
Plot these as
Zeros
Graphing the Tan Function
 What table of values do we use to graph the tan function?
y
x
y
What is the period?


2


4

x
0

4

2








What are the intervals of
increasing or decreasing
for the tan graph?



Domain: x 

2
 k , k  Z
Range:  ,  
Note:
 A function is increasing if it goes up as you travel from left to right
across the graph.
 It is decreasing if the function goes down.


 Does the graph of y = tan(x) have an amplitude?
 “amplitude” is a characteristic of sine and cosine graphs
and it depends on a maximum and a minimum height.
 Since the function y = tan(x) has no maximum or
minimum, it cannot have an amplitude.
y





x















 In the previous unit, we solved first and second degree
trigonometric equations algebraically.
 Such as: cos   cos   1  cos 
2
2
 In this unit, we continue to solve equations algebraically.
 We also use the graphs of trigonometric functions to solve
equations.
 For assessment purposes, you will analyze given graphs to
determine solutions.
 We will also solve trigonometric equations for which the
argument may include a horizontal stretch or a horizontal
translation.


2

1
 Such as: sin  2   2    2
Using graphs to solve trigonometric equations.
 Example 1: Using the graph shown, determine the
general solution for -3sin(2(x-π/4))+3=3
y
y=-3sin(2(x-π/4))+3

x












 Example 2: Using the graph shown, determine the
general solution for 2sin  x  2   1  0
Example 3. Solve
 

4 cos  2  x     1  3
6 
 
for x ε[0,2π]
yy


xx








y=4cos(2(x-π/6))+1



 





Using algebra to solve trigonometric
equations involving horizontal stretches
and translations.
 Lets solve the last 3 problems using algebra
1. Determine the general solution for -3sin(2(x-π/4))+3=3
Solution: First lets simplify what we are solving.

 Let 2(x-π/4)) = m

2 x    k
4

 Solve -3sin(m)+3=3
 x     k
3sin m  0


sin m  0
m  sin1 (0)
m  0, 
m   k ,where k 
4
2
 k  2 k  2 k  
x 


 
2
4
4
4
4

x 
 (2k  1)
4
,where k 
 Remember to check solutions by evaluating each side
of the original equation at all (or a few) of the solution
values.
2. Determine the general solution for

2sin  x    1  0
2

 

4 cos  2  x     1  3 for x  0,2 
6 
 
4. Solve:
3
cos  4  x  900    
, for x  0 ,360 
2
FUN One!
FUN One!
5. Solve: 3sin 4 x  2  2,for x   ,2  Express answers
as exact values, or rounded to 2 decimals.
 Text Page 275 # 1-3, 5, 13a) 14